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A224211 Irregular triangular array read by rows. T(n,k) is the number of n-permutations with exactly k distinct cycle lengths; n>=1, 1<=k<=floor( (-1+(1+8n)^(1/2))/2 ).

%I A224211 #11 May 14 2016 17:22:14
%S A224211 1,1,2,3,6,8,24,50,120,234,120,720,1764,630,5040,11808,7392,40320,
%T A224211 109584,69552,362880,954000,763200,151200,3628800,10628640,8165520,
%U A224211 1330560,39916800,113891040,109010880,25280640,479001600,1486442880,1345687200,381775680,6227020800,18913184640,19773804960,6763236480,87178291200,283465647360,291950568000,102508005600,10897286400
%N A224211 Irregular triangular array read by rows.  T(n,k) is the number of n-permutations with exactly k distinct cycle lengths; n>=1, 1<=k<=floor( (-1+(1+8n)^(1/2))/2 ).
%C A224211 Row sums = A007838.
%H A224211 Alois P. Heinz, <a href="/A224211/b224211.txt">Rows n = 1..300, flattened</a>
%F A224211 E.g.f.: Product_{i>=1} (1 + y*x)^i/i.
%e A224211 :      1;
%e A224211 :      1;
%e A224211 :      2,      3;
%e A224211 :      6,      8;
%e A224211 :     24,     50;
%e A224211 :    120,    234,    120;
%e A224211 :    720,   1764,    630;
%e A224211 :   5040,  11808,   7392;
%e A224211 :  40320, 109584,  69552;
%e A224211 : 362880, 954000, 763200, 151200;
%p A224211 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A224211       `if`(i<1, 0, b(n, i-1)+ (i-1)!*b(n-i, i-1)*
%p A224211       `if`(i>n, 0, binomial(n, i)*x))))
%p A224211     end:
%p A224211 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
%p A224211 seq(T(n), n=1..15);  # _Alois P. Heinz_, Oct 21 2015
%t A224211 nn=15;f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Product[(1+y x^i/i),{i,1,nn}],{x,0,nn}],{x,y}],1]]//Grid
%K A224211 nonn,tabf
%O A224211 1,3
%A A224211 _Geoffrey Critzer_, Apr 01 2013